Mathematics > Commutative Algebra

Title:The Multiplicity Conjecture for Barycentric Subdivisions

Abstract: For a simplicial complex $\Delta$ we study the effect of barycentric
subdivision on ring theoretic invariants of its Stanley-Reisner ring. In
particular, for Stanley-Reisner rings of barycentric subdivisions we verify a
conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of
a standard graded $k$-algebra to the product of the maximal and minimal shifts
in its minimal free resolution up to the height. On the way to proving the
conjecture we develop new and list well known results on behavior of dimension,
Hilbert series, multiplicity, local cohomology, depth and regularity when
passing from the Stanley-Reisner ring of $\Delta$ to the one of its barycentric
subdivision.